Does any of these not work?

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Do these two questions mean the same thing?

[You present several options, wondering if any of those don't work in a given context]
  • 1. Does any of these not work?
  • 2. Doesn't any of these work?

    It seems to me that the position of 'not' might change the perception of the purpose of the question: as if in 1 I'm really interested if any of those don't work, while in 2 I'm more like surprised by the previously given information that none of them works.
 
First of all, use 'do' not 'does'. Second, yes, they'd be used in different ways, with different focus. Third, I'm going to change your example to make it easier to understand.

1) Do any of you not want a piece of cake?

You might use this with a stress on not to ask only those people who don't want a piece of cake to answer you. Your assumption is that there are more people who do want cake than people who don't.

2) Don't any of you want a piece of cake?

This might be used to express surprise or disappointment, for example, that nobody has expressed interest in your cake.
 
First of all, use 'do' not 'does'.
Should I use 'do' even if I believe there's only one thing/person among them that does not?
1) Do any of you not want a piece of cake?
You might use this with a stress on not to ask only those people who don't want a piece of cake to answer you. Your assumption is that there are more people who do want cake than people who don't.
To tell the truth, I didn't get this at all. Sorry. Can we stick to the original example? But I'd like to change it a little bit.

There are three or more options. I think maybe one or two (or even three or more) of them (doesn't) don't work. Which should I ask?
  • Do(es) any of these not work?
  • Do(es)n't any of these work?
There are two options. I think maybe one or two of them (doesn't) don't work. Which should I ask?
  • Do(es) either of these not work?
  • Do(es)n't either of these work?
 
To tell the truth, I didn't get this at all. Sorry.

Really? What didn't you get?

Can we stick to the original example?

Let's use my example, as I made it for a good reason. It's pretty simple—you have a cake and you're cutting it up, wondering who to serve a piece to. Here's a context for 1.

A: Who wants cake?
B: Me.
C: Yes, please.
D: Me too.
E: I do.
F: Yes, please.
G: Thanks.
H: Me!
I: I'd love some.
A: Let me rephrase that—do any of you not want cake?
J: I don't. Thank you.

Now here's a context for 2:

A: Who wants cake?
...
A: Don't any of you want cake? I can't eat it all.
 
Really? What didn't you get?
It's just that I'm interested in my example : ) Want to know which I should use. Could you explain that to me?
 
Give me full context (a short dialogue) to show me clearly what you mean.
 
First of all, I need to thank you that you tried to further explain that to me above.
Give me full context (a short dialogue) to show me clearly what you mean.
Of course! : )

Let's say you're on a learning-a-language forum asking about some alternatives. You have several of them, but you're not sure, you feel like some of them may not work. So, you want to figure out if any of the alternatives do not work. Which would you ask?
  • 1. Do any of these not work?
  • 2. Don't any of these work?
As I wrote above, it seems to me that they are not the same: I think that the position of 'not' might change the perception of the purpose of the question: as if in 1. I'm really interested if any of those don't work, while in 2 I'm more like surprised by the previously given information that none of them works. So, I don't think 2. works in this context.
 
Let's say you're on a learning-a-language forum asking about some alternatives. You have several of them, but you're not sure, you feel like some of them may not work. So, you want to figure out if any of the alternatives do not work. Which would you ask?

Use 1. for that.

Alternately, you could ask : Which of these don't work?

You can't use 2. in this situation, unless you want to show that you're surprised or disappointed, for example, that none of them work.
 
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